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September  2021, 41(9): 4085-4104. doi: 10.3934/dcds.2021029

Best approximation of orbits in iterated function systems

Saisai Shi 1, , Bo Tan 1,, and  Qinglong Zhou 2,

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2. 

School of Science, Wuhan University of Technology, Wuhan 430074, China

* Corresponding author: Bo Tan

Received  October 2020 Revised  December 2020 Published  September 2021 Early access  January 2021

Let
$ \Phi = \{\phi_{i}\colon i\in\Lambda\} $
 be an iterated function system on a compact metric space
$ (X,d) $
, where the index set
$ \Lambda = \{1, 2, \ldots,l\} $
 with
$ l \ge2 $
, or
$ \Lambda = \{1,2,\ldots\} $
. We denote by
$ J $
 the attractor of
$ \Phi $
, and by
$ D $
 the subset of points possessing multiple codings. For any
$ x\in J\backslash D, $
 there is a unique integer sequence
$ \{\omega_{n}(x)\}_{n\geq 1}\subset \Lambda^{\mathbb{N}} $
, called the digit sequence of
$ x, $
 such that
$ \{x\} = \bigcap\limits_n\phi_{\omega_{1}(x)}\circ\cdots\circ\phi_{\omega_{n}(x)}(X). $
In this case we write
$ x = [\omega_{1}(x),\omega_{2}(x),\ldots]. $
 For
$ x, y\in J\backslash D, $
 we define the shortest distance function
$ M_{n}(x,y) $
 as
$ \begin{align*} M_{n}(x,y) = \max\big\{k\in \mathbb{N}\colon \omega_{i+1}(x) = \omega_{i+1}(y),\ldots, &\omega_{i+k}(x) = \omega_{i+k}(y) \; \\&\text{for some}\; 0\leq i \leq n-k\big\}, \end{align*} $
which counts the run length of the longest same block among the first
$ n $
 digits of
$ (x,y). $
In this paper, we are concerned with the asymptotic behaviour of
$ M_{n}(x,y) $
 as
$ n $
 tends to
$ \infty. $
 We calculate the Hausdorff dimensions of the exceptional sets arising from the shortest distance function. As applications, we study the exceptional sets in several concrete systems such as continued fractions system, Lüroth system,
$ N $
-ary system, and triadic Cantor system.
Keywords: Best approximation, Shortest distance function, Iterated function systems.
Mathematics Subject Classification: Primary: 11K55; Secondary: 37F35, 28A80.
Citation: Saisai Shi, Bo Tan, Qinglong Zhou. Best approximation of orbits in iterated function systems. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4085-4104. doi: 10.3934/dcds.2021029
References:
[1]

M. D. Boshernitzan, Quantitative recurrence results, Invent. Math., 113 (1993), 617-631.  doi: 10.1007/bf01244320.  Google Scholar

[2]

D. Bessis, G. Paladin, G. Turchetti and S. Vaienti, Generalized dimensions, entropies, and Liapunov exponents from the pressure function for strange sets, J. Statist. Phys., 51 (1988), 109-134.  doi: 10.1007/bf01015323.  Google Scholar

[3]

L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys., 219 (2001), 443-463.  doi: 10.1007/s002200100427.  Google Scholar

[4]

Y. Bugeaud and B.-W. Wang, Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions, J. Fractal Geom., 1 (2014), 221-241.  doi: 10.4171/jfg/6.  Google Scholar

[5]

I. P. Coornfeld, S. V. Fomin and Ya. G. Sinaǐ, Ergodic Theory, Springer-Verlag, New York, 1982.  Google Scholar

[6]

N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math., 122 (2001), 1-27.  doi: 10.1007/bf02809888.  Google Scholar

[7]

K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, 3$^nd$ edition, John Wiley & Sons, Ltd., Chichester, 2014.  Google Scholar

[8]

J. L. Fernández, M. V. Melián and D. Pestana, Quantitative recurrence properties of expanding maps, preprint, arXiv: math/0703222. Google Scholar

[9]

S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.  doi: 10.4310/mrl.2007.v14.n5.a8.  Google Scholar

[10] G. H. Hardy and E. M. Wright, An Introduction to The Theory of Numbers, 5$^nd$ edition, The Clarendon Press, Oxford University Press, New York, 1979.   Google Scholar
[11]

R. Hill and S. L. Velani, The ergodic theory of shrinking targets, Invent. Math., 119 (1995), 175-198.  doi: 10.1007/bf01245179.  Google Scholar

[12]

R. Hill and S. L. Velani, Metric Diophantine approximation in Julia sets of expanding rational maps, Inst. Hautes Études Sci. Publ. Math., (1997), 193–216. doi: 10.1007/bf02699537.  Google Scholar

[13]

N. Haydn and S. L. Vaienti, The Rényi entropy function and the large deviation of short return times, Ergodic Theory Dynam. Systems., 30 (2010), 159-179.  doi: 10.1017/s0143385709000030.  Google Scholar

[14]

H. Jager and C. de Vroedt, Lüroth series and their ergodic properties, Nederl. Akad. Wetensch. Proc. Ser., 31 (1969), 31-42.  doi: 10.1016/1385-7258(69)90023-7.  Google Scholar

[15]

A. Ya. Khintchine, Continued Fractions, Translated by Peter WynnP. Noordhoff, Ltd., Groningen 1963. doi: 10.1017/s0008439500032033.  Google Scholar

[16]

B. Li, B. W. Wang, J. Wu and J. Xu, The shrinking target problem in the dynamical system of continued fractions, Proc. London Math. Soc., 108 (2014), 159-186.  doi: 10.1112/plms/pdt017.  Google Scholar

[17]

J. Li and X. Yang, On longest matching consecutive subsequence, Int. J. Number Theory., 15 (2019), 1745-1758.  doi: 10.1142/s1793042119500970.  Google Scholar

[18]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc., 73 (1996), 105-154.  doi: 10.1112/plms/s3-73.1.105.  Google Scholar

[19]

R. D. Mauldin and M. Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc., 351 (1999), 4995-5025.  doi: 10.1090/s0002-9947-99-02268-0.  Google Scholar

[20]

D. S. Ornstein and B. Weiss, Entropy and data compression schemes, IEEE Trans. Inform. Theory., 39 (1993), 78-83.  doi: 10.1109/18.179344.  Google Scholar

[21]

L. Peng, On the hitting depth in the dynamical system of continued fractions, Chaos. Solitons. Fractals., 69 (2014), 22-30.  doi: 10.1016/j.chaos.2014.09.003.  Google Scholar

[22]

L. Peng, B. Tan and B. W. Wang, Quantitative Poincaré recurrence in continued fraction dynamical system, Sci. China Math., 55 (2012), 131-140.  doi: 10.1007/s11425-011-4303-9.  Google Scholar

[23]

B. Saussol, Recurrence rate in rapidly mixing dynamical systems, Discrete Contin. Dyn. Syst., 15 (2006), 259-267.  doi: 10.3934/dcds.2006.15.259.  Google Scholar

[24]

B. O. Stratmann and M. Urbánski, Jarník, Julia$\colon$a Diophantine analysis for geometrically finite Kleinian groups with parabolic elements, Math. Scand., 91 (2002), 27-54.  doi: 10.7146/math.scand.a-14377.  Google Scholar

[25]

S. Seuret and B.-W. Wang, Quantitative recurrence properties in conformal iterated function systems, Adv. Math., 280 (2015), 472-505.  doi: 10.1016/j.aim.2015.02.019.  Google Scholar

[26]

B. Tan and B.-W. Wang, Quantitative recurrence properties for beta-dynamical system, Adv. Math., 228 (2011), 2071-2097.  doi: 10.1016/j.aim.2011.06.034.  Google Scholar

[27]

F. Takens and E. Verbitski, Generalized entropies$\colon$Rényi and correlation integral approach, Nonlinearity., 11 (1998), 771-782.  doi: 10.1088/0951-7715/11/4/001.  Google Scholar

[28]

B. Tan and Q. L. Zhou, The relative growth rate for partial quotients in continued fractions, J. Math. Anal. Appl., 478 (2019), 229-235.  doi: 10.1016/j.jmaa.2019.05.029.  Google Scholar

[29]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[30]

Q.-L. Zhou, Dimensions of recurrent sets in $\beta$-symbolic dynamics, J. Math. Anal. Appl., 472 (2019), 1762-1776.  doi: 10.1016/j.jmaa.2018.12.022.  Google Scholar

show all references

References:
[1]

M. D. Boshernitzan, Quantitative recurrence results, Invent. Math., 113 (1993), 617-631.  doi: 10.1007/bf01244320.  Google Scholar

[2]

D. Bessis, G. Paladin, G. Turchetti and S. Vaienti, Generalized dimensions, entropies, and Liapunov exponents from the pressure function for strange sets, J. Statist. Phys., 51 (1988), 109-134.  doi: 10.1007/bf01015323.  Google Scholar

[3]

L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys., 219 (2001), 443-463.  doi: 10.1007/s002200100427.  Google Scholar

[4]

Y. Bugeaud and B.-W. Wang, Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions, J. Fractal Geom., 1 (2014), 221-241.  doi: 10.4171/jfg/6.  Google Scholar

[5]

I. P. Coornfeld, S. V. Fomin and Ya. G. Sinaǐ, Ergodic Theory, Springer-Verlag, New York, 1982.  Google Scholar

[6]

N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math., 122 (2001), 1-27.  doi: 10.1007/bf02809888.  Google Scholar

[7]

K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, 3$^nd$ edition, John Wiley & Sons, Ltd., Chichester, 2014.  Google Scholar

[8]

J. L. Fernández, M. V. Melián and D. Pestana, Quantitative recurrence properties of expanding maps, preprint, arXiv: math/0703222. Google Scholar

[9]

S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.  doi: 10.4310/mrl.2007.v14.n5.a8.  Google Scholar

[10] G. H. Hardy and E. M. Wright, An Introduction to The Theory of Numbers, 5$^nd$ edition, The Clarendon Press, Oxford University Press, New York, 1979.   Google Scholar
[11]

R. Hill and S. L. Velani, The ergodic theory of shrinking targets, Invent. Math., 119 (1995), 175-198.  doi: 10.1007/bf01245179.  Google Scholar

[12]

R. Hill and S. L. Velani, Metric Diophantine approximation in Julia sets of expanding rational maps, Inst. Hautes Études Sci. Publ. Math., (1997), 193–216. doi: 10.1007/bf02699537.  Google Scholar

[13]

N. Haydn and S. L. Vaienti, The Rényi entropy function and the large deviation of short return times, Ergodic Theory Dynam. Systems., 30 (2010), 159-179.  doi: 10.1017/s0143385709000030.  Google Scholar

[14]

H. Jager and C. de Vroedt, Lüroth series and their ergodic properties, Nederl. Akad. Wetensch. Proc. Ser., 31 (1969), 31-42.  doi: 10.1016/1385-7258(69)90023-7.  Google Scholar

[15]

A. Ya. Khintchine, Continued Fractions, Translated by Peter WynnP. Noordhoff, Ltd., Groningen 1963. doi: 10.1017/s0008439500032033.  Google Scholar

[16]

B. Li, B. W. Wang, J. Wu and J. Xu, The shrinking target problem in the dynamical system of continued fractions, Proc. London Math. Soc., 108 (2014), 159-186.  doi: 10.1112/plms/pdt017.  Google Scholar

[17]

J. Li and X. Yang, On longest matching consecutive subsequence, Int. J. Number Theory., 15 (2019), 1745-1758.  doi: 10.1142/s1793042119500970.  Google Scholar

[18]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc., 73 (1996), 105-154.  doi: 10.1112/plms/s3-73.1.105.  Google Scholar

[19]

R. D. Mauldin and M. Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc., 351 (1999), 4995-5025.  doi: 10.1090/s0002-9947-99-02268-0.  Google Scholar

[20]

D. S. Ornstein and B. Weiss, Entropy and data compression schemes, IEEE Trans. Inform. Theory., 39 (1993), 78-83.  doi: 10.1109/18.179344.  Google Scholar

[21]

L. Peng, On the hitting depth in the dynamical system of continued fractions, Chaos. Solitons. Fractals., 69 (2014), 22-30.  doi: 10.1016/j.chaos.2014.09.003.  Google Scholar

[22]

L. Peng, B. Tan and B. W. Wang, Quantitative Poincaré recurrence in continued fraction dynamical system, Sci. China Math., 55 (2012), 131-140.  doi: 10.1007/s11425-011-4303-9.  Google Scholar

[23]

B. Saussol, Recurrence rate in rapidly mixing dynamical systems, Discrete Contin. Dyn. Syst., 15 (2006), 259-267.  doi: 10.3934/dcds.2006.15.259.  Google Scholar

[24]

B. O. Stratmann and M. Urbánski, Jarník, Julia$\colon$a Diophantine analysis for geometrically finite Kleinian groups with parabolic elements, Math. Scand., 91 (2002), 27-54.  doi: 10.7146/math.scand.a-14377.  Google Scholar

[25]

S. Seuret and B.-W. Wang, Quantitative recurrence properties in conformal iterated function systems, Adv. Math., 280 (2015), 472-505.  doi: 10.1016/j.aim.2015.02.019.  Google Scholar

[26]

B. Tan and B.-W. Wang, Quantitative recurrence properties for beta-dynamical system, Adv. Math., 228 (2011), 2071-2097.  doi: 10.1016/j.aim.2011.06.034.  Google Scholar

[27]

F. Takens and E. Verbitski, Generalized entropies$\colon$Rényi and correlation integral approach, Nonlinearity., 11 (1998), 771-782.  doi: 10.1088/0951-7715/11/4/001.  Google Scholar

[28]

B. Tan and Q. L. Zhou, The relative growth rate for partial quotients in continued fractions, J. Math. Anal. Appl., 478 (2019), 229-235.  doi: 10.1016/j.jmaa.2019.05.029.  Google Scholar

[29]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[30]

Q.-L. Zhou, Dimensions of recurrent sets in $\beta$-symbolic dynamics, J. Math. Anal. Appl., 472 (2019), 1762-1776.  doi: 10.1016/j.jmaa.2018.12.022.  Google Scholar

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